The goal of this lecture course is to prove the universal optimality of the E8 and Leech lattices.
This theorem is the main result of a recent preprint "Universal optimality of the E8 and Leech lattices and interpolation formulas" written in collaboration with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Danylo Radchenko. We prove that E8 and Leech lattices minimize energy of every potential function that is a completely monotonic function of squared distance (for example, inverse power laws of Gaussians).
This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. To construct the optimal auxiliary functions attaining these bounds, we prove a new interpolation theorem.
At the last lecture, we will discuss open questions and conjectures which arose from our work.